Neurocomputing 52–54 (2003) 65–71www.elsevier.com/locate/neucom

Burst dynamics under mixed NMDA and AMPAdrive in the models of the lamprey spinal CPG

Alexander Kozlova ;∗ , Anders Lansnera , Sten GrillnerbaStudies of Arti�cial Neural Systems, Department of Numerical Analysis and Computer Science,

Royal Institute of Technology, Stockholm SE-10044, SwedenbNobel Institute for Neurophysiology, Department of Neuroscience, Karolinska Institute,

Stockholm SE-17177, Sweden

Abstract

The spinal CPG of the lamprey is modeled using a chain of nonlinear oscillators. Each oscil-lator represents a small neuron population capable of bursting under mixed NMDA and AMPAdrive. Parameters of the oscillator are derived from detailed conductance-based neuron models.Analysis and simulations of dynamics of a single oscillator, a chain of locally coupled excitatoryoscillators and a chain of two pairs of excitatory and inhibitory oscillators in each segment aredone. The roles of asymmetric couplings and additional rostral drive for generation of a travelingwave with one cycle per chain length in a realistic frequency range are studied.c© 2002 Elsevier Science B.V. All rights reserved.

Keywords: Locomotion; Spinal cord; Lamprey; Modeling

1. Introduction

The lamprey swims by means of body undulations thus forming a quasisinusoidalwave of a single wavelength that travels along the body from head to tail [3]. Thistraveling wave is generated by local oscillators in the range of 0.1–10 Hz with inter-segmental phase lag that is nearly independent of cycle period. Thus for 100 segmentsin the lamprey, the intersegmental phase lag is stabilized around 1% of the cycle. Inexperiments, the brainstem and spinal cord can be maintained in vitro while activityin the locomotor CPG is induced by electric or pharmacological stimulation. Patterns

∗ Corresponding author. Tel.: +46-8-7906903; fax: +46-8-7900930.E-mail address: akozlov@nada.kth.se (A. Kozlov).

0925-2312/03/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.doi:10.1016/S0925-2312(02)00795-6

66 A. Kozlov et al. / Neurocomputing 52–54 (2003) 65–71

emerging during this “Dctive swimming” are similar in their characteristics to the onesobserved in vivo.Our goal in the current study is to suggest through modeling a plausible implemen-

tation of phase and frequency control in the spinal network of lamprey. To this end,we model calcium dynamics in interneurons under mixed NMDA and AMPA driveand explore its consequences for swimming rhythm generation.

2. Description of the cell model

The oscillator equations comprise leakage, NMDA and AMPA inward currents acti-vated synaptically and from the bath, and KCa outward current. Calcium enters the unitthrough NMDA channels. The model is developed by a gradual, step-by-step reductionand simpliDcation of detailed conductance-based models of cells and synapses whichhave been developed earlier [1,2,4,9]. Thus the oscillator model is suGciently simplefor limited mathematical analysis and fast simulations, and still keeps direct correspon-dences of its parameters and state variables to characteristics measured experimentally.This makes this model diHerent from previous simpliDed models [5,7,10].Here, the dynamics of the membrane potential v and the intracellullar calcium con-

centration u is modeled in dimensionless form as

dv=dt = i − v+ ia + in + ik + isa + isn + isg; du=dt = (−u+ icn);ia = a(ea − v); in = np(v)(en − v); ik = ku(ek − v);icn = �np(v)(ecn − v) + �iscn; p(v) = 1=(1 + 0:014e(1−v)=0:12); (1)

where i is the injected current, a and n are the conductances of the bath-activatedAMPA and NMDA channels, k = 4:5 is the conductance of the calcium-dependentpotassium channel, ea = 1; en = 1; ek = −0:14; ecn = 1:29 are the reversal potentials, �is the calcium inKux rate, = 0:016 is the reciprocal of the calcium time constant,and p(v) is the opening function of the voltage-dependent magnesium block. Synapticinput has two excitatory components, isa and isn (AMPA and NMDA, respectively),and one inhibitory component, isg (glycin),

isa = sah(vpre)(ea − v);isn = snh(w)p(v)(en − v); dw=dt = (vpre − w)=�n;isg = sgh(vpre)(eg − v); h(v) = 1=(1 + e(0:3−v)=0:1); (2)

where vpre and v are the pre- and post-synaptic membrane potentials, sa; sn; sg are thesynaptic conductances, ea = 1; en = 1; eg = −0:06 are the reversal potentials, �n is theNMDA receptor activation time constant, and h(v) is the synaptic transfer function.Synapse contribution to calcium inKux is modeled by the current

iscn = snh(w)p(v)(ecn − v): (3)

A. Kozlov et al. / Neurocomputing 52–54 (2003) 65–71 67

v

u

250 ms

β = 0.5

(B)(A)

(D)(C)

0

1

2

3

5

a

4

0 40 50 20 30 10

Oscillations

n

a = 0

0

0.5

1

1.5

β

0 20 30 50 10 40

Oscillations

n

Fig. 1. Single cell dynamics. (A) A cell with recurrent excitation. (B) Typical waveforms of the membranepotential, v, and the intracellular calcium concentration, u. (C, D) Bifurcation diagrams for model (1) withoutrecurrent excitation. Regions of periodic oscillations are shaded.

The cell model (1)–(3) is used for network simulations. All oscillators are identical,only bath drives and synaptic couplings are varied.

3. Results

3.1. Single cell dynamics

The single cell without recurrent excitation is a potential burster under bath NMDAdrive. AMPA application controls the frequency of NMDA-induced oscillations sothe burst frequency gradually increases with the increase of conductance a or theproportional increase of both n and a. The oscillations occur in a limited range of�, as shown in Fig. 1, so the calcium inKux can be neither too high nor toolow.Recurrent excitation increases cell excitability. It shifts the region of oscillations

towards smaller values of n so for the strong enough recurrent excitation the oscillationscan be observed even without bath NMDA drive (not shown).

3.2. Phase locking

Excitatory synapses are able to synchronize nonidentical oscillators for suGcientlystrong reciprocal coupling as shown in Fig. 2A. The oscillator with the higher intrinsic

68 A. Kozlov et al. / Neurocomputing 52–54 (2003) 65–71

(A) (B) sn

0

5

10

20

sa = 1.0 - sn 15

∆ϕ (%

)

0.5 0.6 0.7 0.8 0.9 1

Fig. 2. Synchronization of reciprocally coupled nonidentical oscillators. (A) The left oscillator has 10%more bath AMPA drive. (B) Dependence of the phase diHerence between membrane potentials of the twooscillators, in percents of the cycle duration, on NMDA and AMPA synaptic conductances, sn and sa.

frequency due to the extra bath AMPA drive entrains the slower one so their phasesget locked with the phase of the former leading. The phase diHerence is smaller forstronger couplings and it tends to be larger if the NMDA component dominates asshown in Fig. 2B.

3.3. E-network

A linear chain of oscillators with excitatory couplings to nearest neighbours andrecurrent excitation, the E-network, models the dynamics of a piece of spinal cordsplit along the mid-line (L. Cangiano and S. Grillner, pers. comm.).As shown in Fig. 3 an E-network with 100 symmetrically coupled oscillators gener-

ates forward swimming patterns in the range of 2–6 Hz with intersegmental phase lagbetween 0.8% and 1.3% if an additional AMPA drive is applied to the most rostralsegment of the chain. This implements the ‘trailing oscillator’ hypothesis suggestedearlier [8].

3.4. EI-network

This network consists of two E-networks reciprocally coupled via inhibitory cells asshown in Fig. 4. The EI-network features prominent descending rostro-caudal asymme-try of contralateral inhibitory connections. It is found that this asymmetry is suGcientfor generation of traveling waves even without extra rostral drive as shown in Fig. 5A.Results are veriDed in large-scale simulations [6] as shown in Fig. 5B. This networkmaintains a one cycle per chain (“body”) length phase lag for higher frequencies mosteGciently. In the low frequency range, additional rostral drive helps to correct theintersegmental phase lag.

A. Kozlov et al. / Neurocomputing 52–54 (2003) 65–71 69

(A)

(B) (C)0 10 20 30 40 50

0

1

5

4

3

2

a

1.3%

0.8%

n

1.0%

0 10 20 30 40 50

8

6

4

2

0

0.8%

1.0%

1.3%

Fre

qu

ency

(H

z)

n

Fig. 3. Coordination of intersegmental phase lag in the chain of excitatory oscillators. (A) The most rostraloscillator has 2% more bath AMPA drive. (B) Oscillations occur within the shaded area. Lines in the regionof oscillations show values of n and a corresponding to travelling waves of excitation with intersegmentalphase lags of 0.8%, 1.0%, and 1.3%. Frequency of the traveling waves changes along the lines and variesbetween 2 and 6 Hz (C).

I

I

E

E

Single segment Inhibitory projections Excitatory projections

Fig. 4. ConDguration of EI-network. One excitatory, E, and one inhibitory, I, oscillators build a hemisegment;two symmetric hemisegments comprise a segment. There are 100 segments in the spinal cord. Excitatoryprojections are symmetric, inhibitory projections are 50% weaker in the rostral direction.

4. Conclusion

Chains of coupled nonlinear network oscillators are shown to reproduce impor-tant aspects of the swimming pattern generation in the lamprey spinal CPG. Currents

70 A. Kozlov et al. / Neurocomputing 52–54 (2003) 65–71

Fig. 5. Comparison of oscillations in simpliDed (A) and detailed (B) models of the spinal CPG of thelamprey. Time varies along the horizontal line. Excitation propagates from anterior to posterior along theleft (from segment 0 to hemisegment −100) and the right (from 0 to +100) sides of the spinal cord inout-of-phase manner. Both models generate one cycle per body length.

depending on activation of NMDA receptors and the associated calcium dynamics arekey determinants of the characteristics of these oscillations. Both E- and EI-networksare capable of producing an adequate swimming coordination, with the EI-conDgurationbeing most eGcient.

References

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Alexander Kozlov received his Diploma in Radiophysics and Ph.D. in Physics andMathematics from the University of Nizhny Novgorod, Russia, in 1989 and 1995,respectively. His research in Kungliga Tekniska HSogskolan, Stockholm, includesdynamics of control and pattern generating nervous systems, realistic neuron mod-eling and large-scale simulations.

Anders Lansner received his Ph.D in Computer Science from the Royal Instituteof Technology in 1986. He was appointed Professor in Computer Science in 1999and is manager of the SANS group (Studies of ArtiDcial Neural Systems) foundedin 1987. His research interests range from abstract neural network computationto biophysically detailed computational models of speciDc biological systems andprocesses.

Sten Grillner is a Professor of the Department of Neuroscience, head of the Divi-sion of Neurophysiology and Behavior (The Nobel Institute for Neurophysiology)of Karolinska Institute, Stockholm. His research interest is directed towards anunderstanding of certain aspects of the cellular and network level and how the dif-ferent types of neurons combine to form networks which underlie basic modes ofcoordination such as control of body orientation and locomotion.